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Theorem r19.45av 3014
Description: Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when  A is empty.) (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.45av  |-  ( E. x  e.  A  (
ph  \/  ps )  ->  ( ph  \/  E. x  e.  A  ps ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    A( x)

Proof of Theorem r19.45av
StepHypRef Expression
1 r19.43 3012 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
2 idd 24 . . . 4  |-  ( x  e.  A  ->  ( ph  ->  ph ) )
32rexlimiv 2944 . . 3  |-  ( E. x  e.  A  ph  ->  ph )
43orim1i 517 . 2  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  -> 
( ph  \/  E. x  e.  A  ps )
)
51, 4sylbi 195 1  |-  ( E. x  e.  A  (
ph  \/  ps )  ->  ( ph  \/  E. x  e.  A  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    e. wcel 1762   E.wrex 2810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1592  df-ral 2814  df-rex 2815
This theorem is referenced by: (None)
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